# ACTSC445 Study Guide - Quiz Guide: Likelihood-Ratio Test, Independent And Identically Distributed Random Variables, Random VariableExam

by OC2719308

School

University of WaterlooDepartment

Actuarial ScienceCourse Code

ACTSC445Professor

Jiahua ChenStudy Guide

QuizThis

**preview**shows half of the first page. to view the full**2 pages of the document.**ACTSC 445/845 – Fall 2014

ASSIGNMENT 3 DUE:10:55 am Thursday November 27th

Hand in to Yunran Wei in the TA oﬃce.

Notes:

•You may work on this assignment individually or in groups of up to three people. If

you work in a group, you should submit a single solution with each students’ name

and ID number.

•If you work in a group, each student in the group will receive the same mark.

•There will be no extra credit for students who work individually.

For use in both questions:

Download a data set of weekly returns from Yahoo.com for the period of 1990-2013 for two

stocks, A and B. Use ﬁnancial related industries (banks, insurers, etc). See the note at the

end of the assignment for details of how to do this.

1. For stock A and stock B, separately:

We are interested in studying extreme outcomes of the negative logreturns, per

$100 invested, which is denoted by Xi=−100 log(Si+1/Si) for the ith week, where

Siis your stock price. You may assume that the log-returns are iid with a common

distribution F.

Let Mj,j= 1,2, ..., n denote the maximum values of Xiin the jth block, where each

block is 26 weeks long.

(a) Generate a histogram of the observed Mj.

(b) Using maximum likelihood, estimate the GEV parameters for the distribution.

(c) Compare the data and the ﬁtted model with a Q-Q plot, and comment on the ﬁt.

(d) Use the likelihood ratio test to compare the simpler model, γ= 0 with the model

γ > 0, and comment on the results.

(e) Use sample mean excess plot to ﬁnd an appropriate threshold uso that the ex-

ceedances can be ﬁtted with a generalized Pareto distribution (GPD).

(f) Fit the GPD to the exceedances of the negative logreturns.

(g) Test the hypothesis γ= 0, using the likelihood ratio test.

(h) Based on the GPD you obtained in the previous steps, produce a mean excess

plot of the negative logreturns.

(i) Develop estimates for V aR0.99(X) and CTE0.99(X) by using the PGD you devel-

oped, where Xdenotes the negative logreturn random variable.

(j) Compare the results for Stock A and Stock B.

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